Hands-on Exercise 10

Calibrating Hedonic Pricing Model for Private Highrise Property with GWR Method

1. Overview

To begin with, geographically weighted regression (GWR) is a spatial statistical technique that takes non-stationary variables into consideration (e.g., climate; demographic factors; physical environment characteristics) and models the local relationships between these independent variables and an outcome of interest (also known as dependent variable).

In this hands-on exercise, I will be building hedonic pricing models by using GWR methods. The dependent variable is the resale prices of condominium in 2015. The independent variables are divided into either structural and locational.

1.1 Installing Required Packages

The R packages needed for this exercise are as follows:

• R package for building OLS and performing diagnostics tests

  • olsrr

• R package for calibrating geographical weighted family of models

  • GWmodel

• R package for multivariate data visualisation and analysis

  • corrplot

• Spatial data handling

  • sf

• Attribute data handling

  • tidyverse, especially readr, ggplot2 and dplyr

• Choropleth mapping

  • tmap

pacman::p_load(olsrr, corrplot, ggpubr, sf, spdep, GWmodel, tmap, tidyverse, gtsummary, sfdep)

Installing package into 'C:/Users/Samantha/AppData/Local/R/win-library/4.4'
(as 'lib' is unspecified)

also installing the dependency 'glue'

Warning: unable to access index for repository http://www.stats.ox.ac.uk/pub/RWin/bin/windows/contrib/4.4:
  cannot open URL 'http://www.stats.ox.ac.uk/pub/RWin/bin/windows/contrib/4.4/PACKAGES'

package 'glue' successfully unpacked and MD5 sums checked

Warning: cannot remove prior installation of package 'glue'

Warning in file.copy(savedcopy, lib, recursive = TRUE): problem copying
C:\Users\Samantha\AppData\Local\R\win-library\4.4\00LOCK\glue\libs\x64\glue.dll
to C:\Users\Samantha\AppData\Local\R\win-library\4.4\glue\libs\x64\glue.dll:
Permission denied

Warning: restored 'glue'

package 'gtsummary' successfully unpacked and MD5 sums checked

The downloaded binary packages are in
    C:\Users\Samantha\AppData\Local\Temp\Rtmp06xofg\downloaded_packages

gtsummary installed

Warning in pacman::p_load(olsrr, corrplot, ggpubr, sf, spdep, GWmodel, tmap, : Failed to install/load:
gtsummary

1.2 Importing the Datasets

1.2.1 Aspatial Data

The condo_resale_2015 is in csv file format. The codes chunk below uses read_csv() function of readr package to import condo_resale_2015 into R as a tibble data frame called condo_resale.

condo_resale = read_csv("data/aspatial/Condo_resale_2015.csv")

Rows: 1436 Columns: 23
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (23): LATITUDE, LONGITUDE, POSTCODE, SELLING_PRICE, AREA_SQM, AGE, PROX_...

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

After importing the data file into R, it is important for us to examine if the data file has been imported correctly. By using glimpse(), we can observe that it returns 1,436 rows and 23 columns.

glimpse(condo_resale)

Rows: 1,436
Columns: 23
$ LATITUDE             <dbl> 1.287145, 1.328698, 1.313727, 1.308563, 1.321437,…
$ LONGITUDE            <dbl> 103.7802, 103.8123, 103.7971, 103.8247, 103.9505,…
$ POSTCODE             <dbl> 118635, 288420, 267833, 258380, 467169, 466472, 3…
$ SELLING_PRICE        <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1320…
$ AREA_SQM             <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 168,…
$ AGE                  <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22, 6,…
$ PROX_CBD             <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783402…
$ PROX_CHILDCARE       <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543, 0…
$ PROX_ELDERLYCARE     <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.121…
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.410632,…
$ PROX_HAWKER_MARKET   <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969, 0…
$ PROX_KINDERGARTEN    <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076, 0…
$ PROX_MRT             <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.528…
$ PROX_PARK            <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.116…
$ PROX_PRIMARY_SCH     <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.709…
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.709…
$ PROX_SHOPPING_MALL   <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.307…
$ PROX_SUPERMARKET     <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.581…
$ PROX_BUS_STOP        <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340, 0…
$ NO_Of_UNITS          <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34, 3…
$ FAMILY_FRIENDLY      <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0…
$ FREEHOLD             <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR       <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…

Next, summary() of base R is used to display the summary statistics of cond_resale tibble data frame.

summary(condo_resale)

    LATITUDE       LONGITUDE        POSTCODE      SELLING_PRICE     
 Min.   :1.240   Min.   :103.7   Min.   : 18965   Min.   :  540000  
 1st Qu.:1.309   1st Qu.:103.8   1st Qu.:259849   1st Qu.: 1100000  
 Median :1.328   Median :103.8   Median :469298   Median : 1383222  
 Mean   :1.334   Mean   :103.8   Mean   :440439   Mean   : 1751211  
 3rd Qu.:1.357   3rd Qu.:103.9   3rd Qu.:589486   3rd Qu.: 1950000  
 Max.   :1.454   Max.   :104.0   Max.   :828833   Max.   :18000000  
    AREA_SQM          AGE           PROX_CBD       PROX_CHILDCARE    
 Min.   : 34.0   Min.   : 0.00   Min.   : 0.3869   Min.   :0.004927  
 1st Qu.:103.0   1st Qu.: 5.00   1st Qu.: 5.5574   1st Qu.:0.174481  
 Median :121.0   Median :11.00   Median : 9.3567   Median :0.258135  
 Mean   :136.5   Mean   :12.14   Mean   : 9.3254   Mean   :0.326313  
 3rd Qu.:156.0   3rd Qu.:18.00   3rd Qu.:12.6661   3rd Qu.:0.368293  
 Max.   :619.0   Max.   :37.00   Max.   :19.1804   Max.   :3.465726  
 PROX_ELDERLYCARE  PROX_URA_GROWTH_AREA PROX_HAWKER_MARKET PROX_KINDERGARTEN 
 Min.   :0.05451   Min.   :0.2145       Min.   :0.05182    Min.   :0.004927  
 1st Qu.:0.61254   1st Qu.:3.1643       1st Qu.:0.55245    1st Qu.:0.276345  
 Median :0.94179   Median :4.6186       Median :0.90842    Median :0.413385  
 Mean   :1.05351   Mean   :4.5981       Mean   :1.27987    Mean   :0.458903  
 3rd Qu.:1.35122   3rd Qu.:5.7550       3rd Qu.:1.68578    3rd Qu.:0.578474  
 Max.   :3.94916   Max.   :9.1554       Max.   :5.37435    Max.   :2.229045  
    PROX_MRT         PROX_PARK       PROX_PRIMARY_SCH  PROX_TOP_PRIMARY_SCH
 Min.   :0.05278   Min.   :0.02906   Min.   :0.07711   Min.   :0.07711     
 1st Qu.:0.34646   1st Qu.:0.26211   1st Qu.:0.44024   1st Qu.:1.34451     
 Median :0.57430   Median :0.39926   Median :0.63505   Median :1.88213     
 Mean   :0.67316   Mean   :0.49802   Mean   :0.75471   Mean   :2.27347     
 3rd Qu.:0.84844   3rd Qu.:0.65592   3rd Qu.:0.95104   3rd Qu.:2.90954     
 Max.   :3.48037   Max.   :2.16105   Max.   :3.92899   Max.   :6.74819     
 PROX_SHOPPING_MALL PROX_SUPERMARKET PROX_BUS_STOP       NO_Of_UNITS    
 Min.   :0.0000     Min.   :0.0000   Min.   :0.001595   Min.   :  18.0  
 1st Qu.:0.5258     1st Qu.:0.3695   1st Qu.:0.098356   1st Qu.: 188.8  
 Median :0.9357     Median :0.5687   Median :0.151710   Median : 360.0  
 Mean   :1.0455     Mean   :0.6141   Mean   :0.193974   Mean   : 409.2  
 3rd Qu.:1.3994     3rd Qu.:0.7862   3rd Qu.:0.220466   3rd Qu.: 590.0  
 Max.   :3.4774     Max.   :2.2441   Max.   :2.476639   Max.   :1703.0  
 FAMILY_FRIENDLY     FREEHOLD      LEASEHOLD_99YR  
 Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
 1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000  
 Median :0.0000   Median :0.0000   Median :0.0000  
 Mean   :0.4868   Mean   :0.4227   Mean   :0.4882  
 3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000  
 Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  

1.2.2 Geospatial Data

The geospatial data used in this hands-on exercise is MP14_SUBZONE_WEB_PL. It is in ESRI shapefile format consisting of URA Master Plan 2014’s planning subzone boundaries.

The code chunk below is used to import MP_SUBZONE_WEB_PL shapefile by using st_read() of sf packages.

mpsz = st_read(dsn = "data/geospatial", layer = "MP14_SUBZONE_WEB_PL")

Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `C:\SamanthaxFoo\IS415-GAA\Hands-on_Ex\Hands-on_Ex10\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

Observations

The geometry data in this shapefile consists of polygon features which are used to represent these geographic boundaries. We can also see that the GIS data is in the svy21 projected coordinates system.

1.3 Data Wrangling

1.3.1 Aspatial Data

We first need to convert the condo_resale dibble data frame into a sf object. We will also need to convert the projection from WSG84 into SVY21, which is the projection used in Singapore.

condo_resale.sf <- st_as_sf(condo_resale,
                            coords = c("LONGITUDE", "LATITUDE"),
                            crs=4326) %>%
  st_transform(crs=3414)

Next, head() is used to list the content of condo_resale.sf object.

head(condo_resale.sf)

Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 × 22
  POSTCODE SELLING_PRICE AREA_SQM   AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
     <dbl>         <dbl>    <dbl> <dbl>    <dbl>          <dbl>            <dbl>
1   118635       3000000      309    30     7.94          0.166            2.52 
2   288420       3880000      290    32     6.61          0.280            1.93 
3   267833       3325000      248    33     6.90          0.429            0.502
4   258380       4250000      127     7     4.04          0.395            1.99 
5   467169       1400000      145    28    11.8           0.119            1.12 
6   466472       1320000      139    22    10.3           0.125            0.789
# ℹ 15 more variables: PROX_URA_GROWTH_AREA <dbl>, PROX_HAWKER_MARKET <dbl>,
#   PROX_KINDERGARTEN <dbl>, PROX_MRT <dbl>, PROX_PARK <dbl>,
#   PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
#   PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
#   NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
#   LEASEHOLD_99YR <dbl>, geometry <POINT [m]>

Observations

Notice that st_transform() of sf package has successfully helped us to convert the geometric coordinates from wgs84 (i.e. crs:4326) to svy21 (i.e. crs=3414).

1.3.2 Geospatial Data

Similar to the aspatial data, we need to convert the projection from WSG84 into SVY21.

mpsz_svy21 <- st_transform(mpsz, 3414)

After transforming the projection metadata, we can indeed see that the projection has been transformed to 3414 by using st_crs() of sf package.

st_crs(mpsz_svy21)

Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

Next, we can reveal the extent of mpsz_svy21 by using st_bbox() of sf package.

st_bbox(mpsz_svy21) #view extent

     xmin      ymin      xmax      ymax 
 2667.538 15748.721 56396.440 50256.334 

2. Exploratory Data Analysis (EDA)

2.1 Using Statistical Graphics

We will first plot the distribution of the selling price of the condominiums by using the code chunk below.

ggplot(data=condo_resale.sf, aes(x=`SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

Observations

We can observe that the distribution of the selling price is right-skewed, with a long tail to the right. This skewed distribution is typical of real estate prices, where most of the properties are sold at a lower price, with a few sold at a much higher price.

However, working with the raw selling price can be problematic, especially when the distribution is skewed. We can transform the selling price using the natural logarithm to make the distribution more symmetric.

condo_resale.sf <- condo_resale.sf %>%
  mutate(`LOG_SELLING_PRICE` = log(SELLING_PRICE))

Now let’s plot the distribution of the log-transformed selling price, i.e. LOG_SELLING_PRICE.

ggplot(data=condo_resale.sf, aes(x=`LOG_SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

Observations

From the distrbiution above, we see that the distribution of the log-transformed selling price is more symmetric compared to the raw selling price. This transformation will be useful when we calibrate the hedonic pricing model.

2.2 Multiple Histogram Plots distribution of variables

We will now draw a small multiples of histograms to visualise the distribution of the independent variables in the hedonic pricing model. The code below will create 12 histograms. Then, ggarrange() is used to arrange the histograms in a 3x4 grid.

AREA_SQM <- ggplot(data=condo_resale.sf, aes(x= `AREA_SQM`)) + 
  geom_histogram(bins=20, color="black", fill="light blue")

AGE <- ggplot(data=condo_resale.sf, aes(x= `AGE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_CBD <- ggplot(data=condo_resale.sf, aes(x= `PROX_CBD`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_CHILDCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_CHILDCARE`)) + 
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_ELDERLYCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_ELDERLYCARE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_URA_GROWTH_AREA <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_URA_GROWTH_AREA`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_HAWKER_MARKET <- ggplot(data=condo_resale.sf, aes(x= `PROX_HAWKER_MARKET`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_KINDERGARTEN <- ggplot(data=condo_resale.sf, aes(x= `PROX_KINDERGARTEN`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_MRT <- ggplot(data=condo_resale.sf, aes(x= `PROX_MRT`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_PARK <- ggplot(data=condo_resale.sf, aes(x= `PROX_PARK`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_TOP_PRIMARY_SCH <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_TOP_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

ggarrange(AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, 
          PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT,
          PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH,  
          ncol = 3, nrow = 4)

2.3 Drawing Statistical Point Map

Lastly, we will draw a statistical point map to visualise the distribution of the log-transformed selling price of the condominiums in 2015. The code below will create a statistical point map using the tmap package.

• Notice that tm_dots() is used instead of tm_bubbles().

• set.zoom.limits argument of tm_view() sets the minimum and maximum zoom level to 11 and 14 respectively.

mpsz_svy21 <- st_make_valid(mpsz_svy21)
tmap_mode("view")

tmap mode set to interactive viewing

tm_shape(mpsz_svy21) +
  tm_polygons() +
tm_shape(condo_resale.sf) +  
  tm_dots(col = "SELLING_PRICE",
          alpha = 0.6,
          style = "quantile") +
  tm_view(set.zoom.limits = c(11, 14))

tmap_mode("plot")

tmap mode set to plotting

3. Hedonic Pricing Model

3.1 Simple Linear Regression

We will first set the baseline model using simple linear regression (SLR). The SLR model will be used to estimate the relationship between the log-transformed selling price and the area of the condominium.

The code below will calibrate the SLR model for SELLING_PRICE as the dependent variable and AREA_SQM as the independent variable.

condo.slr <- lm(formula=SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)

The functions summary() and anova() are used to obtain the summary statistics and the ANOVA table of the SLR model, respectively.

summary(condo.slr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3695815  -391764   -87517   258900 13503875 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -258121.1    63517.2  -4.064 5.09e-05 ***
AREA_SQM      14719.0      428.1  34.381  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 942700 on 1434 degrees of freedom
Multiple R-squared:  0.4518,    Adjusted R-squared:  0.4515 
F-statistic:  1182 on 1 and 1434 DF,  p-value: < 2.2e-16

Observations

From the output report above, SELLING_PRICE can be explained using the formula:

\[ y = -258121.1 + 14719x1 \]

The R-squared value of 0.4518 indicates that the area of the condominium can explain 45% of the variation in the selling price.

Since p-value is much smaller than 0.0001, we will reject the null hypothesis that mean is a good estimator of SELLING_PRICE. This will allow us to infer that simple linear regression model above is a good estimator of SELLING_PRICE.

The Coefficients: section of the report reveals that the p-values of both the estimates of the Intercept and ARA_SQM are smaller than 0.001. In view of this, the null hypothesis of the B0 and B1 are equal to 0 will be rejected. As a results, we will be able to infer that the B0 and B1 are good parameter estimates.

To visualise the best fit curve on a scatterplot, we can incorporate lm() as a method function in ggplot’s geometry as shown in the code chunk below.

ggplot(data=condo_resale.sf,  
       aes(x=`AREA_SQM`, y=`SELLING_PRICE`)) +
  geom_point() +
  geom_smooth(method = lm)

`geom_smooth()` using formula = 'y ~ x'

We can tell that there are a few outliers with relatively higher selling prices in the scatterplot.

3.2 Multiple Linear Regression

3.2.1 Visualizing the relationships of the independent variables

Before we begin calibrating the multiple linear regression (MLR) model, we will first visualise the relationships between the independent variables to identify any multicollinearity issues. The code below will create a correlation matrix of the independent variables.

To identify the pattern in the matrix, we also need to consider the order of the variables. There are four methods:

• The “AOE” method is used to order the variables based on the average of the absolute off-diagonal correlations.

• The “FPC” method is used to order the variables based on the first principal component.

• The “hclust” method is used to order the variables based on the hierarchical clustering.

• The “alphabet” method is used to order the variables based on the alphabetical order.

We will use the “AOE” method in this example.

corrplot(cor(condo_resale[, 5:23]), diag = FALSE, order = "AOE",
         tl.pos = "td", tl.cex = 0.5, method = "number", type = "upper")

Observations

From the matrix above, we can clearly see that Freehold is highly correlated to LEASE_99YEAR. This is expected as the two variables are related to the tenure of the property. We will need to remove one of the variables to avoid multicollinearity issues. In this case, we will remove LEASE_99YEAR from the hedonic pricing model.

3.3 Calibrating the Multiple Linear Regression Model

The code chunk below using lm() to calibrate the multiple linear regression model.

condo.mlr <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE    + 
                  PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                  PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + PROX_KINDERGARTEN + 
                  PROX_MRT  + PROX_PARK + PROX_PRIMARY_SCH + 
                  PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
                  PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                data=condo_resale.sf)
summary(condo.mlr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + 
    PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + 
    PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
    PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3475964  -293923   -23069   241043 12260381 

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)           481728.40  121441.01   3.967 7.65e-05 ***
AREA_SQM               12708.32     369.59  34.385  < 2e-16 ***
AGE                   -24440.82    2763.16  -8.845  < 2e-16 ***
PROX_CBD              -78669.78    6768.97 -11.622  < 2e-16 ***
PROX_CHILDCARE       -351617.91  109467.25  -3.212  0.00135 ** 
PROX_ELDERLYCARE      171029.42   42110.51   4.061 5.14e-05 ***
PROX_URA_GROWTH_AREA   38474.53   12523.57   3.072  0.00217 ** 
PROX_HAWKER_MARKET     23746.10   29299.76   0.810  0.41782    
PROX_KINDERGARTEN     147468.99   82668.87   1.784  0.07466 .  
PROX_MRT             -314599.68   57947.44  -5.429 6.66e-08 ***
PROX_PARK             563280.50   66551.68   8.464  < 2e-16 ***
PROX_PRIMARY_SCH      180186.08   65237.95   2.762  0.00582 ** 
PROX_TOP_PRIMARY_SCH    2280.04   20410.43   0.112  0.91107    
PROX_SHOPPING_MALL   -206604.06   42840.60  -4.823 1.57e-06 ***
PROX_SUPERMARKET      -44991.80   77082.64  -0.584  0.55953    
PROX_BUS_STOP         683121.35  138353.28   4.938 8.85e-07 ***
NO_Of_UNITS             -231.18      89.03  -2.597  0.00951 ** 
FAMILY_FRIENDLY       140340.77   47020.55   2.985  0.00289 ** 
FREEHOLD              359913.01   49220.22   7.312 4.38e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 755800 on 1417 degrees of freedom
Multiple R-squared:  0.6518,    Adjusted R-squared:  0.6474 
F-statistic: 147.4 on 18 and 1417 DF,  p-value: < 2.2e-16

3.4 Preparing Publication Quality Table: olsrr method

We can infer that not all the independent variables are significant in explaining the variation in the selling price. To identify the significant variables, we can use the olsrr package to obtain the summary statistics of the MLR model. The olsrr package provides a comprehensive summary statistics of the MLR model, including the ANOVA table, the coefficients, the R-squared value, and the p-values of the estimates.

condo.mlr1 <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                   PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                   PROX_URA_GROWTH_AREA + PROX_MRT  + PROX_PARK + 
                   PROX_PRIMARY_SCH + PROX_SHOPPING_MALL    + PROX_BUS_STOP + 
                   NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
                 data=condo_resale.sf)
ols_regress(condo.mlr1)

                                Model Summary                                 
-----------------------------------------------------------------------------
R                            0.807       RMSE                     751998.679 
R-Squared                    0.651       MSE                571471422208.591 
Adj. R-Squared               0.647       Coef. Var                    43.168 
Pred R-Squared               0.638       AIC                       42966.758 
MAE                     414819.628       SBC                       43051.072 
-----------------------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                                     ANOVA                                       
--------------------------------------------------------------------------------
                    Sum of                                                      
                   Squares          DF         Mean Square       F         Sig. 
--------------------------------------------------------------------------------
Regression    1.512586e+15          14        1.080418e+14    189.059    0.0000 
Residual      8.120609e+14        1421    571471422208.591                      
Total         2.324647e+15        1435                                          
--------------------------------------------------------------------------------

                                               Parameter Estimates                                                
-----------------------------------------------------------------------------------------------------------------
               model           Beta    Std. Error    Std. Beta       t        Sig           lower          upper 
-----------------------------------------------------------------------------------------------------------------
         (Intercept)     527633.222    108183.223                   4.877    0.000     315417.244     739849.200 
            AREA_SQM      12777.523       367.479        0.584     34.771    0.000      12056.663      13498.382 
                 AGE     -24687.739      2754.845       -0.167     -8.962    0.000     -30091.739     -19283.740 
            PROX_CBD     -77131.323      5763.125       -0.263    -13.384    0.000     -88436.469     -65826.176 
      PROX_CHILDCARE    -318472.751    107959.512       -0.084     -2.950    0.003    -530249.889    -106695.613 
    PROX_ELDERLYCARE     185575.623     39901.864        0.090      4.651    0.000     107302.737     263848.510 
PROX_URA_GROWTH_AREA      39163.254     11754.829        0.060      3.332    0.001      16104.571      62221.936 
            PROX_MRT    -294745.107     56916.367       -0.112     -5.179    0.000    -406394.234    -183095.980 
           PROX_PARK     570504.807     65507.029        0.150      8.709    0.000     442003.938     699005.677 
    PROX_PRIMARY_SCH     159856.136     60234.599        0.062      2.654    0.008      41697.849     278014.424 
  PROX_SHOPPING_MALL    -220947.251     36561.832       -0.115     -6.043    0.000    -292668.213    -149226.288 
       PROX_BUS_STOP     682482.221    134513.243        0.134      5.074    0.000     418616.359     946348.082 
         NO_Of_UNITS       -245.480        87.947       -0.053     -2.791    0.005       -418.000        -72.961 
     FAMILY_FRIENDLY     146307.576     46893.021        0.057      3.120    0.002      54320.593     238294.560 
            FREEHOLD     350599.812     48506.485        0.136      7.228    0.000     255447.802     445751.821 
-----------------------------------------------------------------------------------------------------------------

3.5 Preparing Publication Quality Table: gtsummary method

We can also use the gtsummary package to obtain the summary statistics of the MLR model in an elegant and flexible way.

In the code chunk below, tbl_regression() is used to create a well formatted regression report.

tbl_regression(condo.mlr1, intercept = TRUE)

With gtsummary package, model statistics can be included in the report by either appending them to the report table by using add_glance_table() or adding as a table source note by using add_glance_source_note() as shown in the code chunk below.

library(dplyr)
tbl_regression(condo.mlr1, 
               intercept = TRUE) %>% 
  add_glance_source_note(
    label = list(sigma ~ "\U03C3"),
    include = c(r.squared, adj.r.squared, 
                AIC, statistic,
                p.value, sigma))

3.5.1 Checking for multicolinearity

We can also use the olsrr package to check for multicollinearity issues in the MLR model. The olsrr package provides the variance inflation factor (VIF) and the tolerance of the independent variables. The VIF measures the extent of multicollinearity in the model, while the tolerance measures the proportion of the variance of an independent variable that is not explained by the other independent variables.

It provides a collection of very useful methods for building better multiple linear regression models:

• comprehensive regression output

• residual diagnostics

• measures of influence

• heteroskedasticity tests

• collinearity diagnostics

• model fit assessment

• variable contribution assessment

• variable selection procedures

In the code chunk below, the ols_vif_tol() of olsrr package is used to test if there are sign of multicollinearity.

ols_vif_tol(condo.mlr1)

              Variables Tolerance      VIF
1              AREA_SQM 0.8728554 1.145665
2                   AGE 0.7071275 1.414172
3              PROX_CBD 0.6356147 1.573280
4        PROX_CHILDCARE 0.3066019 3.261559
5      PROX_ELDERLYCARE 0.6598479 1.515501
6  PROX_URA_GROWTH_AREA 0.7510311 1.331503
7              PROX_MRT 0.5236090 1.909822
8             PROX_PARK 0.8279261 1.207837
9      PROX_PRIMARY_SCH 0.4524628 2.210126
10   PROX_SHOPPING_MALL 0.6738795 1.483945
11        PROX_BUS_STOP 0.3514118 2.845664
12          NO_Of_UNITS 0.6901036 1.449058
13      FAMILY_FRIENDLY 0.7244157 1.380423
14             FREEHOLD 0.6931163 1.442759

Since the VIF of the independent variables are less than 10. We can safely conclude that there are no sign of multicollinearity among the independent variables.

3.5.2 Test for Non-Linearity

It is also important to test for non-linearity in the model. The ols_plot_resid_fit() of olsrr package is used to test for non-linearity in the model.

ols_plot_resid_fit(condo.mlr1)

From the figure above, we can tell that the residuals are randomly scattered around the zero line. This indicates that there are no signs of non-linearity in the model.

1) Test for Normality Assumption

Lastly, we will test for the normality assumption of the residuals. The ols_plot_resid_hist() of olsrr package is used to test for the normality assumption of the residuals.

ols_plot_resid_hist(condo.mlr1)

The figure above shows that the residuals are normally distributed, which is a key assumption of the multiple linear regression model.

Also, if you want to use a formal statistical test method, you can use the ols_test_normality() of olsrr package to test for the normality assumption of the residuals.

ols_test_normality(condo.mlr1)

Warning in ks.test.default(y, "pnorm", mean(y), sd(y)): ties should not be
present for the one-sample Kolmogorov-Smirnov test

-----------------------------------------------
       Test             Statistic       pvalue  
-----------------------------------------------
Shapiro-Wilk              0.6856         0.0000 
Kolmogorov-Smirnov        0.1366         0.0000 
Cramer-von Mises         121.0768        0.0000 
Anderson-Darling         67.9551         0.0000 
-----------------------------------------------

The summary table above reveals that the p-values of the four tests are way smaller than the alpha value of 0.05. Hence we will reject the null hypothesis and infer that there is statistical evidence that the residual are not normally distributed.

2) Testing for Spatial Autocorrelation

The hedonic model we try to build are using geographically referenced attributes, hence it is also important for us to visual the residual of the hedonic pricing model.

In order to perform spatial autocorrelation test, we need to convert condo_resale.sf from sf data frame into a SpatialPointsDataFrame.

First, we will export the residual of the hedonic pricing model and save it as a data frame.

mlr.output <- as.data.frame(condo.mlr1$residuals)

Next, we will join the newly created data frame with condo_resale.sf object.

condo_resale.res.sf <- cbind(condo_resale.sf, 
                        condo.mlr1$residuals) %>%
rename(`MLR_RES` = `condo.mlr1.residuals`)

Next, we will convert condo_resale.res.sf from simple feature object into a SpatialPointsDataFrame because spdep package can only process sp conformed spatial data objects.

The code chunk below will be used to perform the data conversion process.

condo_resale.sp <- as_Spatial(condo_resale.res.sf)
condo_resale.sp

class       : SpatialPointsDataFrame 
features    : 1436 
extent      : 14940.85, 43352.45, 24765.67, 48382.81  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 23
names       : POSTCODE, SELLING_PRICE, AREA_SQM, AGE,    PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN,    PROX_MRT,   PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, PROX_SHOPPING_MALL, ... 
min values  :    18965,        540000,       34,   0, 0.386916393,    0.004927023,      0.054508623,          0.214539508,        0.051817113,       0.004927023, 0.052779424, 0.029064164,      0.077106132,          0.077106132,                  0, ... 
max values  :   828833,       1.8e+07,      619,  37, 19.18042832,     3.46572633,      3.949157205,           9.15540001,        5.374348075,       2.229045366,  3.48037319,  2.16104919,      3.928989144,          6.748192062,        3.477433767, ... 

Next, we will use tmap package to display the distribution of the residuals on an interactive map.

tmap_mode("view")

tmap mode set to interactive viewing

tm_shape(mpsz_svy21)+
  tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha = 0.4) +
tm_shape(condo_resale.res.sf) +  
  tm_dots(col = "MLR_RES",
          alpha = 0.6,
          style="quantile") +
  tm_view(set.zoom.limits = c(11,14))

Variable(s) "MLR_RES" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.

tmap_mode("plot")

tmap mode set to plotting

The figure above reveal that there is sign of spatial autocorrelation.

To proof that our observation is indeed true, the Moran’s I test will be performed

nb <- dnearneigh(coordinates(condo_resale.sp), 0, 1500, longlat = FALSE)

nb_lw <- nb2listw(nb, style = 'W')

lm.morantest(condo.mlr1, nb_lw)

    Global Moran I for regression residuals

data:  
model: lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT +
PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data = condo_resale.sf)
weights: nb_lw

Moran I statistic standard deviate = 24.366, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Observed Moran I      Expectation         Variance 
    1.438876e-01    -5.487594e-03     3.758259e-05 

The summary table above reveals that the p-value of the Moran’s I test is way smaller than the alpha value of 0.05. Hence we will reject the null hypothesis and infer that there is statistical evidence that the residual are not normally distributed.

Since the Global Moran’s I = 0.1424418 is greater than 0, we can infer that there is sign of positive spatial autocorrelation.

4. Building Hedonic Pricing Models using GWmodel

4.1 Building Fixed Bandwidth GWR Model

4.1.1 Computing the Bandwidth

The first step in calibrating the GWR model is to compute the bandwidth. The bandwidth is a critical parameter in the GWR model as it determines the number of observations that will be used to calibrate the local regression model. Notice that the argument adaptive is set to FALSE indicates that we are interested to compute the fixed bandwidth.

There are several methods to compute the bandwidth, they are: CV cross-validation approach and AIC corrected (AICc) approach. In this example, we will use the cross-validation (CV) method to compute the bandwidth. The CV method is a robust method to compute the bandwidth as it minimizes the prediction error of the GWR model.

bw.fixed <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                     PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA + 
                     PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                     PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                     FAMILY_FRIENDLY + FREEHOLD, 
                   data=condo_resale.sp, 
                   approach="CV", 
                   kernel="gaussian", 
                   adaptive=FALSE, 
                   longlat=FALSE)

Fixed bandwidth: 17660.96 CV score: 8.259118e+14 
Fixed bandwidth: 10917.26 CV score: 7.970454e+14 
Fixed bandwidth: 6749.419 CV score: 7.273273e+14 
Fixed bandwidth: 4173.553 CV score: 6.300006e+14 
Fixed bandwidth: 2581.58 CV score: 5.404958e+14 
Fixed bandwidth: 1597.687 CV score: 4.857515e+14 
Fixed bandwidth: 989.6077 CV score: 4.722431e+14 
Fixed bandwidth: 613.7939 CV score: 1.378294e+16 
Fixed bandwidth: 1221.873 CV score: 4.778717e+14 
Fixed bandwidth: 846.0596 CV score: 4.791629e+14 
Fixed bandwidth: 1078.325 CV score: 4.751406e+14 
Fixed bandwidth: 934.7772 CV score: 4.72518e+14 
Fixed bandwidth: 1023.495 CV score: 4.730305e+14 
Fixed bandwidth: 968.6643 CV score: 4.721317e+14 
Fixed bandwidth: 955.7206 CV score: 4.722072e+14 
Fixed bandwidth: 976.6639 CV score: 4.721387e+14 
Fixed bandwidth: 963.7202 CV score: 4.721484e+14 
Fixed bandwidth: 971.7199 CV score: 4.721293e+14 
Fixed bandwidth: 973.6083 CV score: 4.721309e+14 
Fixed bandwidth: 970.5527 CV score: 4.721295e+14 
Fixed bandwidth: 972.4412 CV score: 4.721296e+14 
Fixed bandwidth: 971.2741 CV score: 4.721292e+14 
Fixed bandwidth: 970.9985 CV score: 4.721293e+14 
Fixed bandwidth: 971.4443 CV score: 4.721292e+14 
Fixed bandwidth: 971.5496 CV score: 4.721293e+14 
Fixed bandwidth: 971.3793 CV score: 4.721292e+14 
Fixed bandwidth: 971.3391 CV score: 4.721292e+14 
Fixed bandwidth: 971.3143 CV score: 4.721292e+14 
Fixed bandwidth: 971.3545 CV score: 4.721292e+14 
Fixed bandwidth: 971.3296 CV score: 4.721292e+14 
Fixed bandwidth: 971.345 CV score: 4.721292e+14 
Fixed bandwidth: 971.3355 CV score: 4.721292e+14 
Fixed bandwidth: 971.3413 CV score: 4.721292e+14 
Fixed bandwidth: 971.3377 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
Fixed bandwidth: 971.3408 CV score: 4.721292e+14 
Fixed bandwidth: 971.3403 CV score: 4.721292e+14 
Fixed bandwidth: 971.3406 CV score: 4.721292e+14 
Fixed bandwidth: 971.3404 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 

The result shows that the recommended bandwidth is 971.3405 metres. This means that the GWR model will use the observations within 971.3405 metres to calibrate the local regression model. Metres is used as the unit of measurement because the data is projected in SVY21.

4.1.2 Calibrating the GWR Model

The next step is to calibrate the GWR model using the recommended bandwidth. The code chunk below will calibrate the GWR model using the recommended bandwidth.

gwr.fixed <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                         PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA + 
                         PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                         PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                         FAMILY_FRIENDLY + FREEHOLD, 
                       data=condo_resale.sp, 
                       bw=bw.fixed, 
                       kernel = 'gaussian', 
                       longlat = FALSE)

The output is saved in a list of class “gwrm”. The code below can be used to display the model output.

gwr.fixed

   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-10-22 18:31:13.865109 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sp, bw = bw.fixed, kernel = "gaussian", 
    longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 971.3405 
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -3.5988e+07 -5.1998e+05  7.6780e+05  1.7412e+06
   AREA_SQM              1.0003e+03  5.2758e+03  7.4740e+03  1.2301e+04
   AGE                  -1.3475e+05 -2.0813e+04 -8.6260e+03 -3.7784e+03
   PROX_CBD             -7.7047e+07 -2.3608e+05 -8.3600e+04  3.4646e+04
   PROX_CHILDCARE       -6.0097e+06 -3.3667e+05 -9.7425e+04  2.9007e+05
   PROX_ELDERLYCARE     -3.5000e+06 -1.5970e+05  3.1971e+04  1.9577e+05
   PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04  7.0749e+04  2.2612e+05
   PROX_MRT             -3.5282e+06 -6.5836e+05 -1.8833e+05  3.6922e+04
   PROX_PARK            -1.2062e+06 -2.1732e+05  3.5383e+04  4.1335e+05
   PROX_PRIMARY_SCH     -2.2695e+07 -1.7066e+05  4.8472e+04  5.1555e+05
   PROX_SHOPPING_MALL   -7.2585e+06 -1.6684e+05 -1.0517e+04  1.5923e+05
   PROX_BUS_STOP        -1.4676e+06 -4.5207e+04  3.7601e+05  1.1664e+06
   NO_Of_UNITS          -1.3170e+03 -2.4822e+02 -3.0846e+01  2.5496e+02
   FAMILY_FRIENDLY      -2.2749e+06 -1.1140e+05  7.6214e+03  1.6107e+05
   FREEHOLD             -9.2067e+06  3.8073e+04  1.5169e+05  3.7528e+05
                             Max.
   Intercept            112793548
   AREA_SQM                 21575
   AGE                     434201
   PROX_CBD               2704596
   PROX_CHILDCARE         1654087
   PROX_ELDERLYCARE      38867814
   PROX_URA_GROWTH_AREA  78515730
   PROX_MRT               3124316
   PROX_PARK             18122425
   PROX_PRIMARY_SCH       4637503
   PROX_SHOPPING_MALL     1529952
   PROX_BUS_STOP         11342182
   NO_Of_UNITS              12907
   FAMILY_FRIENDLY        1720744
   FREEHOLD               6073636
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 438.3804 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6196 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 42263.61 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41632.36 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 42515.71 
   Residual sum of squares: 2.53407e+14 
   R-square value:  0.8909912 
   Adjusted R-square value:  0.8430417 

   ***********************************************************************
   Program stops at: 2024-10-22 18:31:14.738371 

The report shows that the AICc of the gwr is 42263.61 which is significantly smaller than the global multiple linear regression model of 42967.1. This indicates that the GWR model is a better model to explain the variation in the selling price.

4.2 Building Adaptive Bandwidth GWR Model

4.2.1 Computing the Bandwidth

Unlike the fixed bandwidth GWR model, the adaptive bandwidth GWR model does not require the bandwidth to be computed. Instead, the bandwidth is computed for each observation based on the number of observations within a certain distance. The code chunk below will calibrate the adaptive bandwidth GWR model. To do this, we need to set the adaptive argument to TRUE.

bw.adaptive <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE  + 
                        PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE    + 
                        PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                        PROX_PRIMARY_SCH + PROX_SHOPPING_MALL   + PROX_BUS_STOP + 
                        NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                      data=condo_resale.sp, 
                      approach="CV", 
                      kernel="gaussian", 
                      adaptive=TRUE, 
                      longlat=FALSE)

Adaptive bandwidth: 895 CV score: 7.952401e+14 
Adaptive bandwidth: 561 CV score: 7.667364e+14 
Adaptive bandwidth: 354 CV score: 6.953454e+14 
Adaptive bandwidth: 226 CV score: 6.15223e+14 
Adaptive bandwidth: 147 CV score: 5.674373e+14 
Adaptive bandwidth: 98 CV score: 5.426745e+14 
Adaptive bandwidth: 68 CV score: 5.168117e+14 
Adaptive bandwidth: 49 CV score: 4.859631e+14 
Adaptive bandwidth: 37 CV score: 4.646518e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 
Adaptive bandwidth: 25 CV score: 4.430816e+14 
Adaptive bandwidth: 32 CV score: 4.505602e+14 
Adaptive bandwidth: 27 CV score: 4.462172e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 

The result shows that the 30 is the recommended data points to be used.

4.2.2 Calibrating the GWR Model

The code chunk below will calibrate the GWR model using the recommended bandwidth.

gwr.adaptive <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                            PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + 
                            PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                            PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                            NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                          data=condo_resale.sp, bw=bw.adaptive, 
                          kernel = 'gaussian', 
                          adaptive=TRUE, 
                          longlat = FALSE)

Similarly, we can also display the model output.

gwr.adaptive

   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-10-22 18:31:21.146594 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sp, bw = bw.adaptive, kernel = "gaussian", 
    adaptive = TRUE, longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Adaptive bandwidth: 30 (number of nearest neighbours)
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -1.3487e+08 -2.4669e+05  7.7928e+05  1.6194e+06
   AREA_SQM              3.3188e+03  5.6285e+03  7.7825e+03  1.2738e+04
   AGE                  -9.6746e+04 -2.9288e+04 -1.4043e+04 -5.6119e+03
   PROX_CBD             -2.5330e+06 -1.6256e+05 -7.7242e+04  2.6624e+03
   PROX_CHILDCARE       -1.2790e+06 -2.0175e+05  8.7158e+03  3.7778e+05
   PROX_ELDERLYCARE     -1.6212e+06 -9.2050e+04  6.1029e+04  2.8184e+05
   PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04  4.5869e+04  2.4613e+05
   PROX_MRT             -4.3781e+07 -6.7282e+05 -2.2115e+05 -7.4593e+04
   PROX_PARK            -2.9020e+06 -1.6782e+05  1.1601e+05  4.6572e+05
   PROX_PRIMARY_SCH     -8.6418e+05 -1.6627e+05 -7.7853e+03  4.3222e+05
   PROX_SHOPPING_MALL   -1.8272e+06 -1.3175e+05 -1.4049e+04  1.3799e+05
   PROX_BUS_STOP        -2.0579e+06 -7.1461e+04  4.1104e+05  1.2071e+06
   NO_Of_UNITS          -2.1993e+03 -2.3685e+02 -3.4699e+01  1.1657e+02
   FAMILY_FRIENDLY      -5.9879e+05 -5.0927e+04  2.6173e+04  2.2481e+05
   FREEHOLD             -1.6340e+05  4.0765e+04  1.9023e+05  3.7960e+05
                            Max.
   Intercept            18758355
   AREA_SQM                23064
   AGE                     13303
   PROX_CBD             11346650
   PROX_CHILDCARE        2892127
   PROX_ELDERLYCARE      2465671
   PROX_URA_GROWTH_AREA  7384059
   PROX_MRT              1186242
   PROX_PARK             2588497
   PROX_PRIMARY_SCH      3381462
   PROX_SHOPPING_MALL   38038564
   PROX_BUS_STOP        12081592
   NO_Of_UNITS              1010
   FAMILY_FRIENDLY       2072414
   FREEHOLD              1813995
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 350.3088 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 1085.691 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 41982.22 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41546.74 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 41914.08 
   Residual sum of squares: 2.528227e+14 
   R-square value:  0.8912425 
   Adjusted R-square value:  0.8561185 

   ***********************************************************************
   Program stops at: 2024-10-22 18:31:22.472292 

The report shows that the AICc the adaptive distance gwr is 41982.22 which is even smaller than the AICc of the fixed distance gwr of 42263.61.

4.3 Visualizing GWR Output

In addition to regression residuals, the output feature class table includes fields for observed and predicted y values, condition number (cond), Local R2, residuals, and explanatory variable coefficients and standard errors:

• Condition Number: this diagnostic evaluates local colinearity. In the presence of strong local colinearity, results become unstable. Results associated with condition numbers larger than 30, may be unreliable.

• Local R2: these values range between 0.0 and 1.0 and indicate how well the local regression model fits observed y values. Very low values indicate the local model is performing poorly. Mapping the Local R2 values to see where GWR predicts well and where it predicts poorly may provide clues about important variables that may be missing from the regression model.

• Predicted: these are the estimated (or fitted) y values 3. computed by GWR.

• Residuals: to obtain the residual values, the fitted y values are subtracted from the observed y values. Standardized residuals have a mean of zero and a standard deviation of 1. A cold-to-hot rendered map of standardized residuals can be produce by using these values.

• Coefficient Standard Error: these values measure the reliability of each coefficient estimate. Confidence in those estimates are higher when standard errors are small in relation to the actual coefficient values. Large standard errors may indicate problems with local collinearity.

They are all stored in a SpatialPointsDataFrame or SpatialPolygonsDataFrame object integrated with fit.points, GWR coefficient estimates, y value, predicted values, coefficient standard errors and t-values in its “data” slot in an object called SDF of the output list.

4.4 Converting SDF into sf data.frame

To visualize the fields in SDF, we need to first covert it into sf data.frame by using the code chunk below.

condo_resale.sf.adaptive <- st_as_sf(gwr.adaptive$SDF) %>%
  st_transform(crs=3414)

condo_resale.sf.adaptive.svy21 <- st_transform(condo_resale.sf.adaptive, 3414)
condo_resale.sf.adaptive.svy21  

Simple feature collection with 1436 features and 51 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 14940.85 ymin: 24765.67 xmax: 43352.45 ymax: 48382.81
Projected CRS: SVY21 / Singapore TM
First 10 features:
    Intercept  AREA_SQM        AGE  PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
1   2050011.7  9561.892  -9514.634 -120681.9      319266.92       -393417.79
2   1633128.2 16576.853 -58185.479 -149434.2      441102.18        325188.74
3   3433608.2 13091.861 -26707.386 -259397.8     -120116.82        535855.81
4    234358.9 20730.601 -93308.988 2426853.7      480825.28        314783.72
5   2285804.9  6722.836 -17608.018 -316835.5       90764.78       -137384.61
6  -3568877.4  6039.581 -26535.592  327306.1     -152531.19       -700392.85
7  -2874842.4 16843.575 -59166.727 -983577.2     -177810.50       -122384.02
8   2038086.0  6905.135 -17681.897 -285076.6       70259.40        -96012.78
9   1718478.4  9580.703 -14401.128  105803.4     -657698.02       -123276.00
10  3457054.0 14072.011 -31579.884 -234895.4       79961.45        548581.04
   PROX_URA_GROWTH_AREA    PROX_MRT  PROX_PARK PROX_PRIMARY_SCH
1            -159980.20  -299742.96 -172104.47        242668.03
2            -142290.39 -2510522.23  523379.72       1106830.66
3            -253621.21  -936853.28  209099.85        571462.33
4           -2679297.89 -2039479.50 -759153.26       3127477.21
5             303714.81   -44567.05  -10284.62         30413.56
6             -28051.25   733566.47 1511488.92        320878.23
7            1397676.38 -2745430.34  710114.74       1786570.95
8             269368.71   -14552.99   73533.34         53359.73
9            -361974.72  -476785.32 -132067.59        -40128.92
10           -150024.38 -1503835.53  574155.47        108996.67
   PROX_SHOPPING_MALL PROX_BUS_STOP  NO_Of_UNITS FAMILY_FRIENDLY  FREEHOLD
1          300881.390     1210615.4  104.8290640       -9075.370  303955.6
2          -87693.378     1843587.2 -288.3441183      310074.664  396221.3
3         -126732.712     1411924.9   -9.5532945        5949.746  168821.7
4          -29593.342     7225577.5 -161.3551620     1556178.531 1212515.6
5           -7490.586      677577.0   42.2659674       58986.951  328175.2
6          258583.881     1086012.6 -214.3671271      201992.641  471873.1
7         -384251.210     5094060.5   -0.9212521      359659.512  408871.9
8          -39634.902      735767.1   30.1741069       55602.506  347075.0
9          276718.757     2815772.4  675.1615559      -30453.297  503872.8
10        -454726.822     2123557.0  -21.3044311     -100935.586  213324.6
         y    yhat    residual CV_Score Stud_residual Intercept_SE AREA_SQM_SE
1  3000000 2886532   113468.16        0    0.38207013     516105.5    823.2860
2  3880000 3466801   413198.52        0    1.01433140     488083.5    825.2380
3  3325000 3616527  -291527.20        0   -0.83780678     963711.4    988.2240
4  4250000 5435482 -1185481.63        0   -2.84614670     444185.5    617.4007
5  1400000 1388166    11834.26        0    0.03404453    2119620.6   1376.2778
6  1320000 1516702  -196701.94        0   -0.72065800   28572883.7   2348.0091
7  3410000 3266881   143118.77        0    0.41291992     679546.6    893.5893
8  1420000 1431955   -11955.27        0   -0.03033109    2217773.1   1415.2604
9  2025000 1832799   192200.83        0    0.52018109     814281.8    943.8434
10 2550000 2223364   326635.53        0    1.10559735    2410252.0   1271.4073
      AGE_SE PROX_CBD_SE PROX_CHILDCARE_SE PROX_ELDERLYCARE_SE
1   5889.782    37411.22          319111.1           120633.34
2   6226.916    23615.06          299705.3            84546.69
3   6510.236    56103.77          349128.5           129687.07
4   6010.511   469337.41          304965.2           127150.69
5   8180.361   410644.47          698720.6           327371.55
6  14601.909  5272846.47         1141599.8          1653002.19
7   8970.629   346164.20          530101.1           148598.71
8   8661.309   438035.69          742532.8           399221.05
9  11791.208    89148.35          704630.7           329683.30
10  9941.980   173532.77          500976.2           281876.74
   PROX_URA_GROWTH_AREA_SE PROX_MRT_SE PROX_PARK_SE PROX_PRIMARY_SCH_SE
1                 56207.39    185181.3     205499.6            152400.7
2                 76956.50    281133.9     229358.7            165150.7
3                 95774.60    275483.7     314124.3            196662.6
4                470762.12    279877.1     227249.4            240878.9
5                474339.56    363830.0     364580.9            249087.7
6               5496627.21    730453.2    1741712.0            683265.5
7                371692.97    375511.9     297400.9            344602.8
8                517977.91    423155.4     440984.4            261251.2
9                153436.22    285325.4     304998.4            278258.5
10               239182.57    571355.7     599131.8            331284.8
   PROX_SHOPPING_MALL_SE PROX_BUS_STOP_SE NO_Of_UNITS_SE FAMILY_FRIENDLY_SE
1               109268.8         600668.6       218.1258           131474.7
2                98906.8         410222.1       208.9410           114989.1
3               119913.3         464156.7       210.9828           146607.2
4               177104.1         562810.8       361.7767           108726.6
5               301032.9         740922.4       299.5034           160663.7
6              2931208.6        1418333.3       602.5571           331727.0
7               249969.5         821236.4       532.1978           129241.2
8               351634.0         775038.4       338.6777           171895.1
9               289872.7         850095.5       439.9037           220223.4
10              265529.7         631399.2       259.0169           189125.5
   FREEHOLD_SE Intercept_TV AREA_SQM_TV     AGE_TV PROX_CBD_TV
1     115954.0    3.9720784   11.614302  -1.615447 -3.22582173
2     130110.0    3.3460017   20.087361  -9.344188 -6.32792021
3     141031.5    3.5629010   13.247868  -4.102368 -4.62353528
4     138239.1    0.5276150   33.577223 -15.524302  5.17080808
5     210641.1    1.0784029    4.884795  -2.152474 -0.77155660
6     374347.3   -0.1249043    2.572214  -1.817269  0.06207388
7     182216.9   -4.2305303   18.849348  -6.595605 -2.84136028
8     216649.4    0.9189786    4.879056  -2.041481 -0.65080678
9     220473.7    2.1104224   10.150733  -1.221345  1.18682383
10    206346.2    1.4343123   11.068059  -3.176418 -1.35360852
   PROX_CHILDCARE_TV PROX_ELDERLYCARE_TV PROX_URA_GROWTH_AREA_TV PROX_MRT_TV
1         1.00048819          -3.2612693            -2.846248368 -1.61864578
2         1.47178634           3.8462625            -1.848971738 -8.92998600
3        -0.34404755           4.1319138            -2.648105057 -3.40075727
4         1.57665606           2.4756745            -5.691404992 -7.28705261
5         0.12990138          -0.4196596             0.640289855 -0.12249416
6        -0.13361179          -0.4237096            -0.005103357  1.00426206
7        -0.33542751          -0.8235874             3.760298131 -7.31116712
8         0.09462126          -0.2405003             0.520038994 -0.03439159
9        -0.93339393          -0.3739225            -2.359121712 -1.67102293
10        0.15961128           1.9461735            -0.627237944 -2.63204802
   PROX_PARK_TV PROX_PRIMARY_SCH_TV PROX_SHOPPING_MALL_TV PROX_BUS_STOP_TV
1   -0.83749312           1.5923022            2.75358842        2.0154464
2    2.28192684           6.7019454           -0.88662640        4.4941192
3    0.66565951           2.9058009           -1.05686949        3.0419145
4   -3.34061770          12.9836105           -0.16709578       12.8383775
5   -0.02820944           0.1220998           -0.02488294        0.9145046
6    0.86781794           0.4696245            0.08821750        0.7656963
7    2.38773567           5.1844351           -1.53719231        6.2029165
8    0.16674816           0.2042469           -0.11271635        0.9493299
9   -0.43301073          -0.1442145            0.95462153        3.3123012
10   0.95831249           0.3290120           -1.71252687        3.3632555
   NO_Of_UNITS_TV FAMILY_FRIENDLY_TV FREEHOLD_TV  Local_R2
1     0.480589953        -0.06902748    2.621347 0.8846744
2    -1.380026395         2.69655779    3.045280 0.8899773
3    -0.045279967         0.04058290    1.197050 0.8947007
4    -0.446007570        14.31276425    8.771149 0.9073605
5     0.141120178         0.36714544    1.557983 0.9510057
6    -0.355762335         0.60891234    1.260522 0.9247586
7    -0.001731033         2.78285441    2.243875 0.8310458
8     0.089093858         0.32346758    1.602012 0.9463936
9     1.534793921        -0.13828365    2.285410 0.8380365
10   -0.082251138        -0.53369623    1.033819 0.9080753
                    geometry
1  POINT (22085.12 29951.54)
2   POINT (25656.84 34546.2)
3   POINT (23963.99 32890.8)
4  POINT (27044.28 32319.77)
5  POINT (41042.56 33743.64)
6   POINT (39717.04 32943.1)
7   POINT (28419.1 33513.37)
8  POINT (40763.57 33879.61)
9  POINT (23595.63 28884.78)
10 POINT (24586.56 33194.31)

gwr.adaptive.output <- as.data.frame(gwr.adaptive$SDF)
condo_resale.sf.adaptive <- cbind(condo_resale.res.sf, as.matrix(gwr.adaptive.output))

glimpse(condo_resale.sf.adaptive)

Rows: 1,436
Columns: 77
$ POSTCODE                <dbl> 118635, 288420, 267833, 258380, 467169, 466472…
$ SELLING_PRICE           <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ AREA_SQM                <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 1…
$ AGE                     <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22,…
$ PROX_CBD                <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783…
$ PROX_CHILDCARE          <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543…
$ PROX_ELDERLYCARE        <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.…
$ PROX_URA_GROWTH_AREA    <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.4106…
$ PROX_HAWKER_MARKET      <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969…
$ PROX_KINDERGARTEN       <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076…
$ PROX_MRT                <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.…
$ PROX_PARK               <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.…
$ PROX_PRIMARY_SCH        <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.…
$ PROX_TOP_PRIMARY_SCH    <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.…
$ PROX_SHOPPING_MALL      <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.…
$ PROX_SUPERMARKET        <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.…
$ PROX_BUS_STOP           <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340…
$ NO_Of_UNITS             <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34…
$ FAMILY_FRIENDLY         <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0…
$ FREEHOLD                <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR          <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ LOG_SELLING_PRICE       <dbl> 14.91412, 15.17135, 15.01698, 15.26243, 14.151…
$ MLR_RES                 <dbl> -1489099.55, 415494.57, 194129.69, 1088992.71,…
$ Intercept               <dbl> 2050011.67, 1633128.24, 3433608.17, 234358.91,…
$ AREA_SQM.1              <dbl> 9561.892, 16576.853, 13091.861, 20730.601, 672…
$ AGE.1                   <dbl> -9514.634, -58185.479, -26707.386, -93308.988,…
$ PROX_CBD.1              <dbl> -120681.94, -149434.22, -259397.77, 2426853.66…
$ PROX_CHILDCARE.1        <dbl> 319266.925, 441102.177, -120116.816, 480825.28…
$ PROX_ELDERLYCARE.1      <dbl> -393417.795, 325188.741, 535855.806, 314783.72…
$ PROX_URA_GROWTH_AREA.1  <dbl> -159980.203, -142290.389, -253621.206, -267929…
$ PROX_MRT.1              <dbl> -299742.96, -2510522.23, -936853.28, -2039479.…
$ PROX_PARK.1             <dbl> -172104.47, 523379.72, 209099.85, -759153.26, …
$ PROX_PRIMARY_SCH.1      <dbl> 242668.03, 1106830.66, 571462.33, 3127477.21, …
$ PROX_SHOPPING_MALL.1    <dbl> 300881.390, -87693.378, -126732.712, -29593.34…
$ PROX_BUS_STOP.1         <dbl> 1210615.44, 1843587.22, 1411924.90, 7225577.51…
$ NO_Of_UNITS.1           <dbl> 104.8290640, -288.3441183, -9.5532945, -161.35…
$ FAMILY_FRIENDLY.1       <dbl> -9075.370, 310074.664, 5949.746, 1556178.531, …
$ FREEHOLD.1              <dbl> 303955.61, 396221.27, 168821.75, 1212515.58, 3…
$ y                       <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ yhat                    <dbl> 2886531.8, 3466801.5, 3616527.2, 5435481.6, 13…
$ residual                <dbl> 113468.16, 413198.52, -291527.20, -1185481.63,…
$ CV_Score                <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ Stud_residual           <dbl> 0.38207013, 1.01433140, -0.83780678, -2.846146…
$ Intercept_SE            <dbl> 516105.5, 488083.5, 963711.4, 444185.5, 211962…
$ AREA_SQM_SE             <dbl> 823.2860, 825.2380, 988.2240, 617.4007, 1376.2…
$ AGE_SE                  <dbl> 5889.782, 6226.916, 6510.236, 6010.511, 8180.3…
$ PROX_CBD_SE             <dbl> 37411.22, 23615.06, 56103.77, 469337.41, 41064…
$ PROX_CHILDCARE_SE       <dbl> 319111.1, 299705.3, 349128.5, 304965.2, 698720…
$ PROX_ELDERLYCARE_SE     <dbl> 120633.34, 84546.69, 129687.07, 127150.69, 327…
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762.12, 47433…
$ PROX_MRT_SE             <dbl> 185181.3, 281133.9, 275483.7, 279877.1, 363830…
$ PROX_PARK_SE            <dbl> 205499.6, 229358.7, 314124.3, 227249.4, 364580…
$ PROX_PRIMARY_SCH_SE     <dbl> 152400.7, 165150.7, 196662.6, 240878.9, 249087…
$ PROX_SHOPPING_MALL_SE   <dbl> 109268.8, 98906.8, 119913.3, 177104.1, 301032.…
$ PROX_BUS_STOP_SE        <dbl> 600668.6, 410222.1, 464156.7, 562810.8, 740922…
$ NO_Of_UNITS_SE          <dbl> 218.1258, 208.9410, 210.9828, 361.7767, 299.50…
$ FAMILY_FRIENDLY_SE      <dbl> 131474.73, 114989.07, 146607.22, 108726.62, 16…
$ FREEHOLD_SE             <dbl> 115954.0, 130110.0, 141031.5, 138239.1, 210641…
$ Intercept_TV            <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5276150, 1.…
$ AREA_SQM_TV             <dbl> 11.614302, 20.087361, 13.247868, 33.577223, 4.…
$ AGE_TV                  <dbl> -1.6154474, -9.3441881, -4.1023685, -15.524301…
$ PROX_CBD_TV             <dbl> -3.22582173, -6.32792021, -4.62353528, 5.17080…
$ PROX_CHILDCARE_TV       <dbl> 1.000488185, 1.471786337, -0.344047555, 1.5766…
$ PROX_ELDERLYCARE_TV     <dbl> -3.26126929, 3.84626245, 4.13191383, 2.4756745…
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.648105057, -5.6…
$ PROX_MRT_TV             <dbl> -1.61864578, -8.92998600, -3.40075727, -7.2870…
$ PROX_PARK_TV            <dbl> -0.83749312, 2.28192684, 0.66565951, -3.340617…
$ PROX_PRIMARY_SCH_TV     <dbl> 1.59230221, 6.70194543, 2.90580089, 12.9836104…
$ PROX_SHOPPING_MALL_TV   <dbl> 2.753588422, -0.886626400, -1.056869486, -0.16…
$ PROX_BUS_STOP_TV        <dbl> 2.0154464, 4.4941192, 3.0419145, 12.8383775, 0…
$ NO_Of_UNITS_TV          <dbl> 0.480589953, -1.380026395, -0.045279967, -0.44…
$ FAMILY_FRIENDLY_TV      <dbl> -0.06902748, 2.69655779, 0.04058290, 14.312764…
$ FREEHOLD_TV             <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7711485, 1.…
$ Local_R2                <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9073605, 0.…
$ coords.x1               <dbl> 22085.12, 25656.84, 23963.99, 27044.28, 41042.…
$ coords.x2               <dbl> 29951.54, 34546.20, 32890.80, 32319.77, 33743.…
$ geometry                <POINT [m]> POINT (22085.12 29951.54), POINT (25656.…

summary(gwr.adaptive$SDF$yhat)

    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
  171347  1102001  1385528  1751842  1982307 13887901 

4.5 Visualizing local R2

The code chunks below is used to create an interactive point symbol map.

tmap_mode("view")

tmap mode set to interactive viewing

tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "Local_R2",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

tmap_mode("plot")

tmap mode set to plotting

4.6 Visualizing coefficient estimates

The code chunks below is used to create an interactive point symbol map.

tmap_mode("view")

tmap mode set to interactive viewing

AREA_SQM_SE <- tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_SE",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

AREA_SQM_TV <- tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_TV",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

tmap_arrange(AREA_SQM_SE, AREA_SQM_TV, 
             asp=1, ncol=2,
             sync = TRUE)

tmap_mode("plot")

tmap mode set to plotting

Lastly, we can also visualize the GWR output by URA planning region. The code chunk below will visualize the local R2 of the GWR output for the Central region.

tm_shape(mpsz_svy21[mpsz_svy21$REGION_N=="CENTRAL REGION", ])+
  tm_polygons()+
tm_shape(condo_resale.sf.adaptive) + 
  tm_bubbles(col = "Local_R2",
           size = 0.15,
           border.col = "gray60",
           border.lwd = 1)